Distillation Column: a Flexibility Study...Distillation Column: a Flexibility Study Supervisor: prof. flavio manenti Advisors: prof. ludovic montastruc prof. xavier joulia Master Graduation

politecnico di milano

Facoltà di Ingegneria Chimica

Scuola di Ingegneria Industriale e dell’Informazione

Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta"

Master of Science in Chemical Engineering

Distillation Column: a Flexibility Study

Supervisor:

prof . flavio manenti

Advisors:

prof . ludovic montastruc

prof . xavier joulia

Master Graduation Thesis by:

alberto ceschin

Student Id n. 858767

Academic Year 2017-2018

In collaboration with:

institut national polytechnique de toulouse

École Nationale Supérieure des Ingénieurs en Arts Chimiques et Technologiques

Laboratoire de Génie Chimique

Copyright © 2017 – 2018 Alberto CeschinAll rights reserved.

To my family

A B S T R A C T

Throughout history, distillation has been one of the most studied and widespreadseparation method. Nevertheless, due to its high energy inefficiency and new chal-lenges in biorefinery development, radical changes are still needed to meet thedemands of the energy-conscious modern society. In this context, the focus on flex-ibility could reduce energy overconsumption and improve operational feasibilities.Therefore, after having defined flexibility as the ability of a process to operate fea-sibly and optimally in the presence of varying and uncertain conditions, two differ-ent approaches were studied and a new tray-by-tray optimization model, that in-volves both economy and flexibility, was developed. A case studied of a simplifieddebutanizer column inspired from a literature example was rigorously modeled ina C++ programming language, exploiting numerical libraries from BzzMath, andits design was fixed through a standard rigorous economic optimization. Thenfeasibility constrains, that involve hydraulics, heat exchange and purities, were de-fined and the three approaches towards flexibility were applied. The first one wasalready known in literature, together with its role for comparing different configu-rations: once the set of uncertainties and their maximum expected deviations arefixed by the decision maker, this method links operative costs to an adimensionalFlexibility Index and, in addition, it can be used to highlight both bottlenecks andoperability ranges. The second one was developed in this work, combining bothGoal Programming and Linear Scalarization: uncertainties are reduced to flowratesand the Multi Objective Optimization deals both with the maximization of the op-erability region and the minimization of the difference between the average profitsat extreme conditions and the nominal one. Finally, a new flexibility contribute wasimplemented in a standard designing model. The result, gained through a rigor-ous MINLP optimization and with no oversizing factors, confirms both theoreticalknowledge and common experience, increasing the number of tray leads to moreflexibility, but it highlights also the feasibility limits of the obtained configuration.It would be interesting in the future to apply this new design approach to morecomplex configurations, as sequence of columns and extractive distillation.

E S T R AT T O

Storicamente la distillazione è stata una delle tecniche di separazione più studiate ediffuse. Nonostante ciò, a causa della sua alta inefficienza energetica e delle nuovesfide nel campo delle bioraffinerie, servono ancora cambiamenti radicali per incon-trare le necessità della società odierna, più attenta ai consumi energetici. Quindiin questo lavoro, dopo aver definito la flessibilità come la capacità di un processodi accomodare ottimamente variazioni e incertezze, sono state sviluppati due di-versi approcci al suo studio ed è stato definito un nuovo modello di ottimizzazioneche ne tenga conto già in fase di progettazione. Successivamente un caso studiodi una colonna debutanatrice è stato modellato rigorosamente, usando il linguag-gio di programmazione C++ e avvalendosi della libreria numerica BzzMath, e lesue variabili di progetto sono state fissate mediante ottimizzazione economica inbase alle condizioni nominali. Una volta definiti anche i vincoli, che comprendonoidraulica, scambio termico e specifiche sulla purezza, al caso studio sono state ap-plicate le metodologie precedentemente trattate. La prima era già stata definita inletteratura e applicata al confronto di più processi per evincerne il più flessibile:consiste nello scegliere un insieme di variabili incerte, assieme alle loro massimedeviazioni attese, e nell’attribuire un Indice di Flessibilità, correlato ai costi op-erativi, alla configurazione. In aggiunta sono stati evidenziati anche estremi delprocesso e possibili costi operativi per ogni valore ammissibile di flessibilità. Laseconda è stata invece sviluppata in questo lavoro, combinando Programmazionea Obiettivi assieme a Scalarizzazione Lineare: le incertezze sono ridotte di numeroai flussi molari entranti e l’ottimizzazione multi obbiettivo massimizza i range dioperabilità minimizzando la differenza tra profitto medio e profitto nominale. In-fine, un nuovo contributo, che sintetizza in modo attivo il concetto di flessibilità,è stato aggiunto alla procedura standard che combina costi operativi e costo delcapitale iniziale per l’ottimizzazione del progetto di una colonna. Il risultato, ot-tenuto tramite ottimizzazione di variabili sia continue che discrete e senza fattori disovradimensionamento, non solo conferma le aspettative, ovvero che aumentandoil numero di piatti si ottiene una maggior flessibilità della colonna, ma definisceanche i limiti del processo. Sarebbe interessante in futuro applicare quest’ultimaprocedura di progettazione a configurazioni più complesse come treni di colonneo distillazioni estrattive, poichè risultato potrebbe essere non del tutto scontato.

C O N T E N T S

Abstract viiEstratto ixList of Figures xiiiList of Tables xvNomenclature xvi1 introduction 1

1.1 Distillation and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 flexibility in distillation columns 5

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 A Flexibility Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 A New Flexibility Analysis . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Combining Flexibility with Design . . . . . . . . . . . . . . . . . . . . 16

2.5 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 case study 19

3.1 Debutanizer column . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Initial temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Distillation column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Diameter Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 optimal design 33

4.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Result, comparison with a simulation software and discussion . . . . 36

5 feasibility constrains 41

5.1 Flooding Constrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Weeping Constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Duty Constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 flexibility index 47

6.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7 flexibility analysis 53

7.1 Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.4 Additional Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8 flexible design 61

8.1 Flexibility Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8.3 Approximation of Vertices . . . . . . . . . . . . . . . . . . . . . . . . . 63

9 conclusion and future developments 69

bibliography 71

L I S T O F F I G U R E S

Figure 1.1 Conventional vs smart design . . . . . . . . . . . . . . . . . . 3

Figure 2.1 Trade-off curves of cost vs. flexibility for alternative designconfigurations. Choice 2 is more expensive than 1 at nomi-nal condition, but is cheaper at greater Flexibility Index . . . 7

Figure 2.2 Largest hyper-rectangle expected in the space of uncertain-ties. Maximum expected variations are highlighted, in bothdirections and for each nominal uncertainties. Only 2 di-mensions are reported . . . . . . . . . . . . . . . . . . . . . . 8

Figure 2.3 In red a hypothetical feasible domain R around the nominalvalue N of uncertainties . . . . . . . . . . . . . . . . . . . . . 9

Figure 2.4 In blue the scaled hyper-rectangle in the space of uncertain-ties. In black the one made up of the maximum expectedvariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2.5 Scaled hyper-rectangles for δ = 0.33, 0.66, 1 and 1.33 in thespace of uncertainties . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.6 Largest hyper-rectangle in the space of uncertainties, hencehere δ = F.I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.7 Non convex feasible region: solutions may not lie on a vertex 13

Figure 2.8 In red the vertices used for evaluating the profit of the blackhyper-rectangle, around the nominal point N. In blue theaspiration level for flexibility . . . . . . . . . . . . . . . . . . 17

Figure 3.1 Conventional distillation column, with partial reboiler andtotal condenser . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.2 Adiabatic flash at fixed pressure . . . . . . . . . . . . . . . . 24

Figure 3.3 Tray i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 3.4 Feed tray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 3.5 Partial reboiler, tray 0 . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 3.6 Condenser, tray Ns . . . . . . . . . . . . . . . . . . . . . . . . 27

Figure 3.7 Sparsity pattern of the Jacobian matrix. Partial condenser,21 trays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 4.1 Minimization of TAC $/yr . . . . . . . . . . . . . . . . . . . 37

Figure 4.2 Temperature Profile inside the column . . . . . . . . . . . . . 38

Figure 4.3 Molar flow rate profiles inside the column . . . . . . . . . . 38

Figure 4.4 Molar fraction profiles inside the column . . . . . . . . . . . 39

Figure 4.5 Molar fraction profiles inside the column . . . . . . . . . . . 39

Figure 5.1 Comparison between vapor velcity in the critical tray andflooding velocity at increasing flow rate. All the other vari-ables are fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 5.2 Performance diagram: simplified hydraulics of a tray. Areaof satisfactory operation is shaded . . . . . . . . . . . . . . . 44

Figure 5.3 Tray efficiencies across a broad range of loadings . . . . . . 44

Figure 6.1 Two possible alternatives of revamping an old column . . . 50

Figure 6.2 Debutanizer column: Flexibility vs Opex. The Flexibility In-dex is equal to 0.57. Light blue: flooding constrain violated.Red point: duty constrains violeted . . . . . . . . . . . . . . . 50

Figure 6.3 Flexibility vs OPEX, maximum expected operative costs inthe range of feasibility . . . . . . . . . . . . . . . . . . . . . . 52

Figure 7.1 Flexibility analysis problem in two dimensions. The firstaspiration level is represented by the blue rectangle, com-posed by the expected deviations, whereas the second as-piration level is represented by an average profit of vertices(red points) as close as possible to the profit of the nominalvalue N of uncertainties . . . . . . . . . . . . . . . . . . . . . 54

Figure 7.2 Feasible iterations. Flexibility goal is ψ1F1; profit goal isψ2F2. Dotted line is for indicating the minimum . . . . . . . 56

Figure 7.3 Upper and lower profile of propylene and propane. Dottedlines for minimum and boundaries, traced ones for startingvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 7.4 Upper and lower profile of butane and pentane. Dotted linesfor minimum and boundaries, traced ones for starting values 58

Figure 8.1 Values of contributes and their sum during the optimization 64

Figure 8.2 Upper and lower profile of propylene and propane duringthe optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 8.3 Upper and lower profile of n-butane and n-pentane duringthe optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 8.4 Number of trays during the optimization . . . . . . . . . . . 65

Figure 8.5 The points for each side are increased, in order to see howgood was the approximation of considering only vertices forevaluating the average OPEX. This is done for the solutionof the previous optimization . . . . . . . . . . . . . . . . . . . 67

L I S T O F TA B L E S

Table 3.1 Inlet compositions and outlet specifications . . . . . . . . . . 20

Table 3.2 Heat Capacity Coefficients of Gases . . . . . . . . . . . . . . 23

Table 3.3 Thermodynamic properties . . . . . . . . . . . . . . . . . . . 23

Table 3.4 SRK binary coefficients . . . . . . . . . . . . . . . . . . . . . . 23

Table 3.5 Design Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 4.1 Overall Heat-Transfer Coefficients in Refinery Services . . . 35

Table 4.2 Utilities costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Table 4.3 Results and comparison with Aspen Hysis . . . . . . . . . . 37

Table 4.4 Results of FUG method. No efficiency is taken in account . . 40

Table 5.1 Comparison of deck trays . . . . . . . . . . . . . . . . . . . . 43

Table 5.2 Nominal and limit values for duties . . . . . . . . . . . . . . 45

Table 6.1 Uncertainties considered. Nominal values and maximumexpected deviations . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 6.2 First combinations of critical uncertainties for which flood-ing constrain is violated . . . . . . . . . . . . . . . . . . . . . 49

Table 6.3 First combinations of critical uncertainties for which dutyconstrains is violated . . . . . . . . . . . . . . . . . . . . . . . 51

Table 7.1 Starting values, upper and lower Boundaries. . . . . . . . . 53

Table 7.2 Results of the Flexibility analysis for ψ1 = 1, ψ2 = 1E2.Molar flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Table 7.3 Results of the Flexibility analysis for ψ1 = 1, ψ2 = 1E2.Functions’ values . . . . . . . . . . . . . . . . . . . . . . . . . 58

Table 7.4 Results of the Flexibility Analysis for ψ1 = 1., ψ2 = 1.E2.Molar flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Table 7.5 Results of the Flexibility Analysis for ψ1 = 1., ψ2 = 1.E4.Values of functions . . . . . . . . . . . . . . . . . . . . . . . . 59

Table 8.1 Variables of the optimization. Upper and lower boundariesand starting values . . . . . . . . . . . . . . . . . . . . . . . . 62

Table 8.2 Parameters for Linear Scalarization . . . . . . . . . . . . . . . 62

Table 8.3 Results of the Flexibility analysis. Molar flow rates . . . . . . 63

Table 8.4 Function values at minimum . . . . . . . . . . . . . . . . . . 66

Table 8.5 Results and comparison with old optimal design . . . . . . . 66

N O M E N C L AT U R E

latin variable unit

A Area m2

C Cost $

Cp Heat capacity at constant pressure J/K/kmol

D Diameter of the column m

F Feed molar flowrate kmol/hr

FI Flexibility Index

H Spacing between trays m

HE Heat exchanger

h Enthalpy J/kmol

L Liquid molar flowrate kmol/hr

Ns Total number of stages i

Nc Total number of components j

Nw Total number of vertices w

K K-factor

k Index for uncertainties, from 1 to p

P Pressure bar

p Total number of uncertainties

OT Operative Time s

Q Duty flow J/hr

R Ideal gas constant J/K/mol

SF Derating factor

T Temperature K

U Overall Heat-Transfer Coefficient Btu/◦F/hr/ft2

u Velocity m2

V Vapor molar flowrate kmol/hr

w Vertex index

x Liquid molar fraction kmol/mol

y Vapor molar fraction kmol/mol

Z Compressibility factor

z Feed molar fraction kmol/mol

superscript mean

i Stage number

l Liquid phase

N Nominal value for an uncertainty

R Residual propriety

v Vapor phase

* Referred to ideal gas

+ Positive deviation

- Negative deviation

° Volume basis 1/m3

subscript mean

b Bottom distillate

c Critical propriety

con Referred to condenser

d Top distillate

DC Down-comer

feed Referred to feed

g Guess value

j Species number j

k Uncertainty number k

L Lower boundary

max Maximum allowed

N Nominal value

net Net

opt Optimal

reb Referred to reboiler

SB

U Upper boundary

w Vertex number w

greek variable unit

α Relative volatility -

∆ Maximum expected deviation -

δ Deviation -

ε Small quantity -

θ Uncertainty

ρ Density kg/m3

σ Surface tension dynes/cm

φ Fugacity coefficient in a mixture

ψ Scalar for Linear Scalarization -

1I N T R O D U C T I O N

"Civilization begins with distillation."

Faulkner , W. 1

1.1 distillation and efficiency

Distillation is a thermal process for separating mixtures of two or more substancesinto their component fractions of desired purity, based on differences in volatil-ity of components, by application and removal of heat. At the commercial scale,distillation has many applications, from the separation of crude oil into fractions(gasoline, diesel, kerosene, etc.) to the water purification and desalination, from thesplitting of air into its components (oxygen, nitrogen and argon), to the distillationof fermented solutions or the production of distilled beverages with higher alcoholcontent.

The art of distillation is believed to have originated in China (∼ 800 BC) withearly applications in the production of alcoholic beverages and the concentrationof essential oils from natural products (Kiss, 2013a). Nevertheless distillation un-derwent enormous development only during last century due to the petrochemicalindustry and as such it is one of the most important technologies in the global en-ergy supply system. Currently every transportation fuel goes through at least onedistillation column on its way from oil to readily usable fuel and, if all industrialsectors are considered, there are hundreds of thousands distillation columns in op-eration all around the world. Also considering the switch to renewable sources ofenergy such as biomass, the most likely transportation fuel will be ethanol, butanolor derivatives. The synthesis of these products leads typically to highly aqueousmixtures that require distillation to separate fuel from water and consequently itremains the separation method of choice in the chemical process industry (Kiss,2013b).

But there is one main issue with distillation. If products and feed streams aretaken in account, it is clear that the separation by distillation requires a decreaseof entropy (∆SDistillation < 0), so it cannot happen spontaneously. An additionof heat is used in practice to make the process thermodynamically possible. If the

1 Nobel Prize in literature (1897-1962). It has to be interpreted as an idiosyncratically baroque sentence,not as a truth.

2 introduction

conventional distillation is taken in account, high-level energy is fed at the baseof the column and about the same amount of energy is released at the top, unfor-tunately at a much lower temperature level (Soave, 2012). Practically, this cause adecrease of thermodynamic efficiency to values as 18% for air separation units, 12%for crude oil units and only 5% for ethylene and propylene production (Haselden,1958).

The quantity of energy increases drastically considering biorefinery processes.Butanol, in ABE production via fermentation for example, is considered a superiorbio-fuel for its characteristics similar to petro-fuel (energy density, low water mis-cibility, flammability and corrosiveness) but requires huge quantities of heat dueto its very low concentration (less than 3 wt%) in the fermentation broth (Patrascuand Bîldea, 2017). If the worldwide goal is reducing the CO2 emission, maybeswitching from an oil-based economy to an bio-based one, this issue has to beovertaken. Some tricks are already used to reduce energy costs, such as heat in-tegration and process intensification. A modern refinery is the example of theextreme grade of optimization that nowadays is reached.

Moreover an issue that bothers distillation and pushes further this energy con-sumption is the presence of non-idealities in the mixture: from the non smoothbehavior of the liquid-vapor curve till the presence of azeotropes. Petrochemicalindustries have learned how to deal with them, but still there are new challenges inthe world of biorefineries. Costs of separations are still too demanding in term ofenergy separation, reaching values of twice the energy recovered from fuel in caseof butanol from ABEW (Siirola, 1996). Finding a better model and the optimumseparation’s path can be very challenging. In this framework extractive distillation,liquid-liquid separation, fractional-adsorption or pressure swing distillation aresometimes mandatory and sometimes better alternatives to conventional distilla-tion. But still conventional distillation is the starting point when a new challengein homogeneous liquid separation is facing on, despite all the energy it requires.

Therefore efficiency is a key word in this context. It has to be improved fromthe early stages considering the long life of a distillation column (typically around20 years), using a ’smart’ and ’green’ column design that involves typically threelevels (Summers and Philling, 2012).

• Process design and configuration: the pursuit is to achieve the highest yieldwith the least amount of energy. The solution is use sequence of multiplecolumns or modifying the process configuration.

• Construction materials and resources: minimize the use of utilities and con-struction materials, getting closer to the mechanicals needs. Nowadays tow-ers have smallest diameters and lowest practical height, whereas in the pastthey used to have a spare capacity (Gorak and Sorensen, 2014).

• Internal design optimization: improving transport and physical propertiesexploiting new internals.

1.2 flexibility 3

Figure 1.1: Conventional vs smart design

1.2 flexibility

In the framework of reducing energy requirements, it becomes more important toconsider the entire life of a column at the design stage. Nominal conditions andspecifications can change in time. Pushing the operating condition to new ones,which the column was not designed for, can cost a lot in term of resources. Em-blematic is again the case of biorefineries, where the broth changes from batch tobatch and the optimal design solution at nominal conditions could not be the moreflexible one.

Dealing with conventional distillation, the obvious response would be to in-crease the number of theoretical stages (fig. 1.1). This will reduce the reflux ratiofor the nominal case but will increase the starting price of the column (CapitalCost, CAPEX).

Indeed the ultimate goal of the work is motivating these choices, in other wordsjustifying, on an economical base and with a case study, the need of modifying thedesign of a column when uncertainties are present. A rigorous distillation modelwill be used to legitimize empirical over-design factors and rules of thumb. Butfirst it will be fundamental to start with an existing column; so literature trackswill be followed to define a Flexibility Index (F.I.), related to costs, that can be usedto compare different distillation solutions, to identify the range of operative costsand to locate primary bottlenecks. Secondly, flexibility will be looked at from areverse and new point of view, that is trying to optimize uncertainties on a profitbase. Finally, a standard design procedure will be revisited, with all the knowl-edge learned from the experiences in previous problems, and flexibility will beintegrated in the procedure.

2F L E X I B I L I T Y I N D I S T I L L AT I O N C O L U M N S

2.1 definition

Flexibility is the ability of a process to operate feasibly and optimally in the pres-ence of varying and uncertain conditions. Every process in industry have to ad-dress this problem, but this work is focused on multicomponent distillation in asingle conventional column. Variables in this framework are many and mainly:

• Operational parameters. They depends from the plant upsets, market condi-tions, feed stock supply and they represent the design flexibility which onedesires to build the process into. For example:

– Product specifications

– Value of products

– Inlet temperature

– Inlet composition

– Inlet pressure

– Inlet flow-rates

• Design uncertainties. They came from lack of experimental data or goodmodels and they represent uncertainties which need to be protected against,causing the process to operate infeasible (i.e. not meeting the specifications).

– Heat transfer coefficients, foul-ing factor

– Column coefficients (efficiency,K-value)

– Cooling water temperature

– Maximum allowed vapor veloc-ity

Traditionally engineers have looked for flexibility by designing the distillationcolumn at the nominal (steady state-most likely) value of the parameters, and thenapplying some empirical over-design factors and introducing large storage andsurge tanks for raw materials, intermediates and products. But this approach doesnot give much insight into the degree of flexibility and the amount of over-sizingmust be difficult to justify on economic grounds.

Moreover, once the over-sized process is built, it can be interesting to investigatehow much the operative conditions can be pushed far from the designed case andat which cost this operation can be possible. If the answer can be clear for a process

6 flexibility in distillation columns

engineer, it is not always so immediate for a non-skilled operator or for a manageron the economic field. Have an instrument that can give clearly an insight into thecapabilities of the process would improve the dialog between different sides of theplant.

In particular, one scenario gives a new sense to this ancient problem: in a biore-finery environment, where the produced fermentation broth has always a differentcomposition, water content is huge and many azeotropes are present, it is very im-portant to know where the limits of the process lay and which are the costs thathave to be paid, if the separation is moving away from the nominal value.

But investigating the flexibility of the distillation of a complex mixture is not aneasy task. Articles in literature always address mixtures close to ideality and solvenon rigorous distillation models. Usually it is preferred to use an approximatemethod to reduce the complexity of the problem, such as the Fenske-Underwood-Gilliland one (Stichlmair and Fair, 1998), or non accurate thermodynamic models,derived from hypothesis like perfect gases or ideal liquids. In these cases it is pos-sible to focus more on the mathematical background and robust and systematicsolutions are possible (Paules and Floudas, 1988 and Wagler and Douglas, 1988).But these solutions are still very sensitive to all parameters and are not reliable,considered the simplicity of the model used.

A second very popular approach to solve a flexibility problem is doing a sensi-tivity analysis in a dynamic software environment and moving with one variableat time away from the nominal operating values. This approach is more reliableand gives strong details about the process. But the number of variables it can traitis very limited and it becomes extremely time consuming if the goal is analyzingall the possible combinations to avoid unpredictable unfeasible points and hugerises of OPEX.

Since ad-hoc scenarios and over-design empirical factors will never verify sys-tematically and sturdily the flexibility of a design, hence the choice done in thiswork is to study the flexibility of a designed column in two totally different ways:

• First, correlating flexibility to an economic index, in order to plot a curveFlexibility vs Operative Costs (OPEX), that starts from the nominal optimumvalue, varies when uncertainties increase and stops when the column reachesthe first bottleneck.

• Second, finding the optimal hypervolume of operability in the space of un-certainties.

Both the problems deal with steady state conditions: no dynamic behavior andcontrollability are taken in account. Many are the attempts in literature to definea Flexibility Index for a process, but the second approach to trait the flexibilityproblem is new and its efficacy will be discussed.

2.2 a flexibility index 7

Figure 2.1: Trade-off curves of cost vs. flexibility for alternative design configurations.Choice 2 is more expensive than 1 at nominal condition, but is cheaper atgreater Flexibility Index

Moreover, in the last part of this work, an attempt to introduce flexibility at thedesign stage is done. For simplicity only one design variable will be consideredin the optimization, the number of theoretical plates, but this will be enough toexplain the methodology and the aspects that require a further investigation.

2.2 a flexibility index

Defining a Flexibility Index for a chemical process is a known problem, alreadyaddressed by many authors in the past. One of the first complete dissertation wasdone by Swaney and Grossman, 1985. It deals with all the chemical processes ingeneral, so distillation too, even if it is not directly mentioned. The basic motivationbehind this work was the need of an instrument to compare different process con-figurations at the same conditions, evaluating if the less expensive configurationat the nominal conditions would have been still the best one far from them. Forthis reason arises the need of defining a Flexibility Index that can relate operativecosts to flexibility, in order to compare different process configurations at variousconditions (fig. 2.1). Moreover, with this instrument it is possible to visualize whenthe first bottleneck occurs and have an insight of the maximum operative cost ateach value of the flexibility index.

The first step is to define the set of p uncertain parameters (θk, with k = 1, ..p)that will vary: the process will have to accommodate these variations through its


Figure 2.2: Largest hyper-rectangle expected in the space of uncertainties. Maximum ex-pected variations are highlighted, in both directions and for each nominal un-certainties. Only 2 dimensions are reported

degree of freedom z, control variables; these for a conventional distillation columnare the operating variables which can be manipulated during operations, as refluxrate, cooling water flow rate, steam temperature, ... Each parameter θk should beindependent with a nominal value (θNk ) at the design stage, that will be consid-ered as starting point; if it is true that uncertain parameters are not necessary in-dependent, it is also true that it is always possible to correlate them with algebraicfunctions based on truly independent parameters. Then two expected variationsfor each parameter, one in negative side, one in positive side, have to be chosen.This decision is very sensitive, since all the results will depend on it, and requiresa detailed knowledge of the process, statistical data and empirical rules. Moreover,negative and positive variations are generally different, but in this work they areassumed to be equal. Once defined these two variations for each parameters, ∆θ+kand ∆θ−k , a region of operation will be described in the space of uncertain variables.In two dimensions this equals to the construction of a rectangle (fig. 2.2), whereasof a hyper-rectangle in more dimensions.

In this space of uncertainties, once the design is fixed, a domain R will exist (fig.2.3), made up of all the points that can guarantee feasible operation to the process.Moreover, for each one of these points it is possible to evaluate the operative costand then a profit. The shape of this set of points is very peculiar to the process con-sidered and will depend on its model: the vector of equations (h) and the vector ofconstrains (g):


Figure 2.3: In red a hypothetical feasible domain R around the nominal value N of uncer-tainties.

h(d, z, θ, x) = 0g(d, z, θ, x) 6 0

Where d is the vector of the design variables (fixed, once the process is built),z is the vector of the control variables that represents the degrees of freedom foraccommodating the uncertainties θ and x is the subset of the remaining variableshaving the same dimension as h. Equations in the model of a distillation columnare typically the MESH ones: material balances, equilibrium relationships, summa-tion equations and heat (enthalpy) balances; on the contrary constrains are aboutpurities, hydraulics and heat exchangers. It is possible to re-write the equalities asimplicit functions of x and then use this relations in the constrains to have reducedconstrains in function of only d, z and θ.

h (d, z, θ, x) = 0 → x (d, z, θ) = x

And with substitution:

g (d, z, θ, x (d, z, θ)) = f (d, z, θ) 6 0


Figure 2.4: In blue the scaled hyper-rectangle in the space of uncertainties. In black theone made up of the maximum expected variations

Now it is possible to define the feasible region R:

R = {θ | [∃z | f(d, z, θ) 6 0]}

Since the goal is comparing together all the variables to find the first combina-tion that will result in a bottleneck, it is useful to adimensionalize deviations ofuncertainties with respect their maximum variation.

δ+k =θk − θ

Nk

∆θ+k, δ−k =

θNk − θk

∆θ−kwith: k = 1..p

In this way, it will be possible to describe the hyper-rectangle T in the space ofuncertainties around the nominal point N (fig. 2.4), by simply varying δk (fig. 2.5).

The Flexibility Index (FI) is represented by the largest hyper-rectangle T that canbe expanded around the nominal point in the feasible region (fig. 2.6). The charac-teristic of this rectangle is that its sides are proportional to the expected deviations(i.e.: δk is the same for all the uncertainties).

T (δ) ={θ |(θN − δ∆θ−

)6 θ 6

(θN + δ∆θ+

)}, with: δ > 0


Figure 2.5: Scaled hyper-rectangles for δ = 0.33, 0.66, 1 and 1.33 in the space of uncer-tainties

Figure 2.6: Largest hyper-rectangle in the space of uncertainties, hence here δ = F.I.


In other words, the Flexibility Index of a distillation column is the maximumallowable value for δ that will include only feasible points in the hyper-rectangleT . Hence the mathematical problem is set as follows:

FI = max δ (2.1)

with : maxθ∈T(δ)

minzmaxj∈J

fj(d, z, θ) 6 0

T (δ) ={θ |(θN − δ∆θ−

)6 θ 6

(θN + δ∆θ+

)}

Where FI is the maximum deviation of uncertain parameters from the nominalvalues for which feasible operation can be guarantee by proper manipulation ofthe control variables z. The object of chapter 6 will be finding this maximum ad-missible value δ for a distillation column case study. Moreover, it will be plottedthe Flexibility Index vs OPEX, in order to have the maximum cost that has to bepaid for each value of flexibility. It have to be noticed that the solution of problem2.1 is generally very difficult, since it implies the exploration of all the space ofuncertainties around the nominal values until a bottleneck is reached. If this oper-ation can be easily done with a couple of uncertainties, it become extremely timeconsuming if the number arises.

To avoid this problem, the solution proposed in literature is to analyze only ver-tices of this hyper-rectangle T , but it is necessary to study first the feasibility regionR, in order to assure that the solution lies on a vertex as in fig. 2.6.

A different process with a different feasibility region R is considered now as anexample. Looking at fig. 2.7, it is possible to notice that if one analyzes only thevertices of the dotted hyper-rectangle T ′, he would estimate wrongly the FlexibilityIndex. In fact T ′ includes points that are not feasible at all and the right FlexibilityIndex is represented by hyper-rectangle T .

After this example, the most obvious requirement that comes up in mind is toassure that the feasible region R is convex. By the way this condition is too de-manding, since even in a non convex domain the solution could lies on a vertex.

Therefore an analytical study is required. Details can be found in Swaney andGrossman, 1985, here only the conclusion will be reported. Only for the specialcase in which the constrain functions f(d, z, θ) are:

1. quasi convex in z and

2. one dimensional quasi convex in θ

2.3 a new flexibility analysis 13

Figure 2.7: Non convex feasible region: solutions may not lie on a vertex

the solution of the problem 2.1 will lie at a vertex of the hyper-rectangle T . Asa consequence, it will be possible to analyze only these vertices, neglecting sidesand hence internal points of the hyper-rectangle. In particular, given a number pof uncertainties, the algorithm will have to analyze 2p vertices.

It is noticeable that the number of vertices increases exponentially and there-fore can still be very huge. Two algorithms can be proposed to reduce further thisnumber: a direct search one (based on local linearization for limiting the searchin all the possible combinations) and an implicit one (based on the branch andbound technique and on the strong hypothesis that all the constraints function aremonotonic in their uncertain parameters). In this work will be preferred to ana-lyze systematically every vertices, in order to plot the final hyper-rectangle T inthe graph Flexibility vs OPEX [$].

2.3 a new flexibility analysis

In Ramos, Gourmelon, and Montastruc, 2017, flexibility is studied from a differentpoint of view. Again at the beginning an optimal design for nominal condition isfound, but then a new flexibility analysis is carried out. The set of uncertaintiesis made of the molar flowrates of the feed, but they are no more input variable,but unknowns. Indeed they are optimized in order to minimize first, and thenmaximize, the recovery of one product, according to feasibility constrains. A stepfurther can be finding the whole hyper-space, hence increasing independently ver-tices, until a bottleneck is reached or the variations are equal to their maximum


expected values. But it should be better to introduce also some economic constrainor directly a quantity in the object function, to avoid the exploration in poorly con-venient regions.

Indeed the second method to study flexibility is based on Goal Programming(Miettinen, 1999), a possible approach to Multiobjective Optimization. Here theidea is to find:

1. the biggest feasible hyper-rectangle in the space of uncertainties

2. with an average profit as close as possible tothe profit at nominal values ofuncertainties.

In this kind of optimization the decision maker is asked to specify aspirationlevels for the objective functions; these aspiration levels are assumed to be selectedso that they are not achievable simultaneously. Then, deviations from these aspira-tion levels are minimized. An objective function jointly with an aspiration level isreferred as a goal. This methodology is now examined and will be applied to thecase study in chapter 7.

As it was anticipated, the first goal deals with maximizing the flexibility; hencethere is the need of fixing the maximum hyper-rectangle in the space of uncertainparameters, and this will be the first aspiration level. It is the decision maker thatfixes for each nominal value of p uncertainties (θNk with k = 1..p) a minimum anda maximum expected value (θNk −∆θ−k and θNk +∆θ+k ). The process will have to op-erate feasibly in an inner region, as close as possible to this large hyper-rectangle.So basically the variables of this Multiobjective Optimization are the lower value(θ−k = θNk − (δ−k · ∆θ

−k )) and the upper value (θ+k = θNk + (δ+k · ∆θ

+k )) of each un-

certainty p; but, differently from section 2.2, now the deviations (δ+or−k ) are nomore proportional among each others but they are independent. The difference involume of the hyper-rectangles can be computed and will be the quantity to beminimize. It has to be observed that, since deviations are different, it is better toadimensionalize them according to the maximum expected; in this way the secondquantity, the one that refers to the largest hyper-rectangle, will be equal to one. Thefunction can be written as follow:

Min F1 =Min

(Πk

( (θNk + δ+k

)−(θNk − δ−k

)(θNk +∆θ+k

)−(θNk −∆θ−k

))− 1

)2

=Min

(Πk

(θ+k − θ−k∆θ+k −∆θ−k

)− 1

)2with: k = 1...p

Subject to: hydraulics and heat exchange constrains

2.3 a new flexibility analysis 15

The second goal concerns the minimization of the difference between the aver-age value of the profit in the hyper-rectangle and the one at the nominal valueof uncertainties (point N). To do so, the profit of the hyper-rectangle has to bedefined. Since again it is not possible to analyze all the points inside the hyper-rectangle, an approximation will be carried out: profit will be evaluated only atevery vertex w (Nw is the total number of vertices, 2p). The sum of all these prof-its, if weighted according to the number of vertices, ideally is the average profit ofthe hyper-rectangle. Moreover, since it is assumed that bottlenecks occur only atvertices, once the feasibility is guarantee for vertices, it will be assure for all the in-ternal points. A better choice would have been to discretize the region according toa grid and evaluate profits in the middle of each hyper-cell, after having checkedthat all vertices are feasible. A convergence analysis then could give a precioussight of how much wrong, the estimate of taking only vertices, is.

Min F2 =

∑w

(Profitw − ProfitN)

Nw · ProfitN

2with: w = 1...NwSubject to: hydraulics and heat exchange constrains

Hence the object of the problem is to minimize both F1 and F2. The most straight-forward method for handling Multi-Objective optimization is to formulate a single-composite objective function with different scalars (weights ψi) associated witheach one of the single objective functions. This method is called Linear Scalariza-tion (Miettinen, 1999).

Min F1, F2 =Min (ψ1F1 +ψ2F2)

The composite objective function is the following:

Min (F1, F2) =

=Min

ψ1(Πk


)− 1

)2+ψ2

∑w


Nw · ProfitN

2

with: w = 1...Nw,

k = 1...p



Notwithstanding, now the weights ψi have to be fixed. The more rigorous wayis to fix to 1 their sum and then optimize all the combinations, so that differentweights are used to generate Pareto optimal solutions (the decision maker finallywill chose the most satisfactory one). In order to avoid many optimizations andcarry out only one, a different approach is used (a similar one is reported in Ramos,Herreros, and Gómez, 2013). The choice of weights is left to the decision maker, ac-cording to the following iterative logic:

1. Arbitrary weights are chosen, and the optimization problem was partly solved.

2. If the solution appears to be enough smooth (and a minimum is found), theweights chosen are assumed to be valid.

3. Otherwise, the weights are adjusted by increasing or decreasing them in anarbitrary way according t the results obtained.

It is possible to visualize the problem in two dimensions in fig. 2.8. The vari-ables that define the space of uncertainties are θ1 and θ2; the nominal values ofthese variables are θN1 and θN2 . The goal is to find the biggest black hyper-rectangle(hence the values for θ−1 , θ+1 , θ−2 and θ+2 ) around the nominal point N, ideally upto the blue rectangle constituted by the maximum deviations expected of uncer-tainties. This task might not be possible since feasibility constrains exist. Each redpoint is a vertex, a combination of two different variables to be optimized. Theaverage value of the profit of a single iteration (black square) is the sum of all theprofits at the vertices, the red points, divided by the number of vertices, here 4since there are only 2 variables.

Chapter 7 is entirely dedicated to the application of this method to the casestudy and to the discussion of results.

2.4 combining flexibility with design

A final attempt to combine flexibility with design has been made. The design vari-able chosen is the number of plates of the column, and a new function has to bedefined. The standard one for the rigorous design of a distillation column is theminimization of Total Annual Cost (TAC): the sum of both Capital Costs (CAPEX)and Operative Costs (OPEX) (Kiss, 2013a).1 But this function doesn’t take in ac-count flexibility, because usually it is based on a nominal case of uncertainties.

TAC [$] =CAPEX

payback period+ OPEX

1 More details in Chapter 4

2.4 combining flexibility with design 17

Figure 2.8: In red the vertices used for evaluating the profit of the black hyper-rectangle,around the nominal point N. In blue the aspiration level for flexibility

In Wagler and Douglas, 1988 it is suggested to optimize not the nominal pointbut a defined number of "near critical points". That means the designer has tochoose some combinations of parameters in the uncertainties region that are notnecessary only the vertices of the feasible region, but a set of combinations use-ful to describe the uncertainties space in which the column has to be designed. Ofcourse a feasibility check is mandatory for each point and the function to minimizeis a Linear Scalarization of the cost C at each point i. In this way the designer cangive a different importance (i.e. weight) to each near critical point and it is possibleto ensure the minimum cost of the column for a fixed number of critical points.

Min F =Min∑i

ψiCi

where∑i

ψi = 1

A different perspective is given by Hoch and Eliceche, 1996. Here the Index ofFlexibility, therefore all the set of uncertainties with their nominal values and ex-pected maximum variations, is set and the design has to accommodate that givenIndex of Flexibility, hence to assure feasible operation for that hyper-rectangle.


But a more appreciate goal would be not fixing the uncertainty space but leav-ing the optimizer free to find the best one inside a maximum expected region. Themain idea is to combine the concept of flexibility together with minimization ofcosts. Many attempts were done and at the end the most reliable answer is aboutintroducing the flexibility quantity from the previous paragraph in a modified TACfunction, involving both Goal Programming and Linear Scalarization:

Min (TAC, F1) =

=Min

ψ3 CAPEXpayback period

+ψ4

∑wOPEXw

Nw+ψ5

(Πk


)− 1

)2with: w = 1...Nw,

k = 1...p


Basically the cheapest configuration has to accommodate the largest variationof uncertainties, according to their maximum expected deviations and feasibilityconstrains. From this point of view the nominal point loses its importance in OPEXevaluation, in favor of an arithmetic average of OPEX at vertices. The concept ofProfit was not used because the procedure for finding the price depends to muchfrom the decision maker. Other design variables (such as feed plate, the columndiameter and the areas of the heat exchangers) are evaluated according to the nom-inal case and to the number of plates of each iteration, according to the standardprocedure explained in Chapter 4 that minimizes duty at the reboiler. Accordingto this, a design solution, so a single different CAPEX value, will correspond toevery value of the number of plates.

2.5 tools

A general issue with these different problems is the huge quantity of simulationsneeded to explore all the possible combinations of the set of variables investigated.Moreover sometimes not all the variables are continuous, as the feed plate and thenumber of plates, and commercial simulator softwares cannot menage optimiza-tion with them. This means dealing with Mixed Integer non Linear Programming(MINLP), still very challenging our days. Hence it has been chosen to model thedistillation column in a C++ programming language exploiting the solvers fromBzzMath 7.2 library (Buzzi-Ferraris and Manenti, 2015). In this library parallel com-puting is already implemented, so all the cpu is used when the program runs. Acomparison with the result modeled in the commercial software Aspen HYSYS®will be showed.

3C A S E S T U D Y

The case study is now introduced and the methodologies anticipated in Chapter 2

will be applied in Chapter 6, 7 and 8.

3.1 debutanizer column

The example chosen is the modified debutanizar column taken from the article ofHoch, Eliceche, and Grossmann, 1995, that resembles the real debutanizer presentin almost all the refineries (fig. 3.1). The aim of this unit is usually to separate thec4 fraction from heavier c5 and c6+ fractions, but in this work the separation isfurther simplified: only n-butane and n-pentane are taken in account, with smallerimpurities of propylene and propane, as it is possible to see in tab. table 3.1. Thereason behind the choice of this case study is the quasi ideality of the separationand the availability of the data from the paper. Moreover, it was already proved intat article that for this problem the solution lies at a vertex of the hyper-rectangle T .

Two specifications are given, hence the first goal is to find the best number oftrays and the feed location, since duties and reflux ratio are assumed as controlvariables. To do so, a cost function is minimized. The equations behind the columnmodel are the MESH ones and, since the mixtures is composed by light hydrocar-bons at low pressure (about 4 bar), Eos SRK with binary interaction parametersis chosen. By analyzing the flow rates it is possible to find the proper diameterand by knowing the duties it is possible to roughly evaluate the areas of the heatexchangers. These two values, together with number of plates, are necessary toestimate the cost of the column and so carrying the optimization process on. Thecolumn is made of sieve trays since it is the cheapest and most spread solution.

Feed is liquid at bubble point (15 bar), whereas the column operates at constantpressure (4 bar). Stages are adiabatic and thermodynamic equilibrium is limiteddue to a Murphree efficiency factor of 0.9.

20 case study

Figure 3.1: Conventional distillation column, with partial reboiler and total condenser

feed bottom product top product

[kmol/hr] [mol/mol] [mol/mol]

Propane 0.2

Propylene 0.2

Butane 24 0.96

Pentane 12 0.98

Table 3.1: Inlet compositions and outlet specifications

3.2 thermodynamics

The mixture can be considered almost ideal, but since accurate data of the opera-tion will be required in the next sections for the evaluation of the flexibility indexand the analysis of flexibility, it is not possible to simplify too much the model ofthe column. Therefore, SRK equation with binary interaction parameters is used topredict vapour-liquid equilibrium and residual enthalpy values. All the equationsused and reported below are from (Rota, 2015).

Z3 −Z2 +(A−B−B2

)Z−AB = 0 (3.1)

3.2 thermodynamics 21

Where: A =

aP

R2T2

B =bP

RT

a =(∑Nc

j=1 xja0.5j

)2b =∑Ncj xibj

Aj =

ajPj

R2T2j

Bj =bjPj

RTj

aj = 0.42748

(RTCj

)2PCj

kj

bj = 0.08664RTCj

PCj

kj =(1+ Sj

(1− T0.5

R,j

))2

Sj = 0.48+ 1.574ωj − 0.176ω2j

Where Nc is the number of j components: propylene, propane, n-butane andn-pentane. Definition of reduced properties is the following:

Pred,j =P

Pc,j

22 case study

Tred,j =T

Tc,j

The two roots of the equation (3.1) are then found and discerned: one for liquidphase, the other for vapor phase. For both of them residual enthalpy and fugacitycoefficients are computed.

hR = RT

(Z− 1−

A

B(1+ E) ln

(Z+B

Z

))

φj = exp

(Bj

B(Z− 1) +

A

B

(Bj

B− 2

√Aj

A

)ln

(Z+B

Z

)− ln(Z−B)

)with: j = 1..Nc

Where:

E =

∑Ncj=1 xjSj

(ajTRj

kj

)0.5

a0.5

Enthalpy is the sum of ideal gas phase and residual one.

h =∑j

h∗j + hR =∑j

∫TTref

Cpj (T)dT + hR

Where:

Cpj (T) = Aj +BjT +CjT2 +DjT

3 + EjT4 [J/mol/K]

Since the evaluation of the equilibrium is needed many time, in the code wasdeveloped an ad-hoc class that receive the values of compositions, temperatureand pressure and gives back the vectors of liquid and vapor compositions withrespective fugacity coefficients (φj), compressibility factor (Z), and molar enthalpyof the liquid and vapor streams (h). The integral is solved analytically since theheat capacity is a polynomial. All the necessary parameters are taken from Yaws,2003, Perry and Green, 2007 and Aspen HYSYS® and reported from table 3.2 to

3.3 initial temperature 23

species a b c d e

Propylene 31.298 0.07245 1.9481E-04 -2.158E-07 6.2974E-11

Propane 28.277 0.11600 1.9597E-04 -2.327E-07 6.8669E-11

nButane 20.056 0.28153 -1.3143E-05 -9.457E-08 3.4149E-11

nPentane 26.671 0.32324 4.2820E-05 -1.664E-07 5.6036E-11

Table 3.2: Heat Capacity Coefficients of Gases

species TC PC ω

[K] [bar]

Propylene 365.0 46.2041 0.1455

Propane 369.9 42.5666 0.1532

n-Butane 425.2 37.9662 0.2008

n-Pentane 469.6 33.7512 0.2522

Table 3.3: Thermodynamic properties

table 3.4. The reference temperature (Tref) is 298.15 K. Roots of equation (3.1) arecomputed analytically without the use of a numerical library.

3.3 initial temperature

The first step is evaluating the temperature of the feed F, since it is known that isliquid at bubble condition. The vector of composition (z0) is given, so the equationof boiling point is used. The Key factor K0j used is the ratio between the fugacitycoefficients and has to be evaluated for each species j.

∑j

z0jK0j = 1

K0j =φl,0j

φv,0j

with: j=1..Nc

i j ki,j i j ki,j

propylene propane 0.0075 propane n-butane 0.00082

propylene n-butane 0.00163 propane n pentane 0.0027

propylene n-pentane 0.00406 n-butane n-pentane 0.00055

Table 3.4: SRK binary coefficients

24 case study

Figure 3.2: Adiabatic flash at fixed pressure

The solver used is part of the BzzNonLinearSystem class (Buzzi-Ferraris andManenti, 2014).

3.4 flash

Since entering in the column there is a pressure drop, from 15 bar to 4 bar, it iswell recommended to divide the flash problem from the distillation one for numer-ical reasons (Kister, 1992). An adiabatic flash at fixed pressure (fig. 3.2) has to besimulated to evaluate the state of the two phases stream. Flash equations in thiscase are in order: four molar balances, four equilibrium relationships, one enthalpybalance and two stoichiometric equations.

F0z0j = F

vzvj + Flzlj

zljφlj = z

vj φvj

F0h0 = Flhl + Fvhv∑j

zlj = 1∑j

zvj = 1

where j=1..Nc

Fugacity coefficients and enthalpies are given by the ad-hoc SRK routine. Againthe solver used belongs to the BzzNonLinearSystem class (Buzzi-Ferraris and Ma-nenti, 2014).

3.5 distillation column

In Hoch, Eliceche, and Grossmann, 1995, the distillation column is a conventionalone, with total condenser and partial reboiler (fig. 3.1). Stages are numbered bot-tom to top, from the stage 0 that is the partial reboiler to the total condenser, stage

3.5 distillation column 25

Figure 3.3: Tray i

N. The equations needed are again molar balances, equilibrium correlations, en-ergy balances and stoichiometric ones. Internal stages are adiabatic.

Equations for tray i (fig. 3.3) are:

Li+1xi+1j − Lixij + Vi−1yi−1j − Viyij = 0

Li+1hl,i+1 − Lihl,i + Vi−1hv,i−1 − Vihv,i = 0

xijφl,ij = yijφ

v,ij∑

j

xij = 1∑j

yij = 1

where:

j = 1..Nc

i ∈ [1..ifeed − 1]∩ [ifeed + 1..Ns− 1]

Equations for the feed stage (fig. 3.4) are below, where i is equal to the positionof the tray.

Flzlj + Fvzvj + L

i+1xi+1j − Lixij + Vi−1yi−1j − Viyij = 0

Flhlfeed + Fvhvfeed + L

i+1hl,i+1 − Lihl,i + Vi−1hv,i−1 − Vihv,i = 0

xijφl,ij = yijφ

v,ij∑

j

xij = 1∑j

yij = 1

where: j = 1..Nc

26 case study

Figure 3.4: Feed tray

Figure 3.5: Partial reboiler, tray 0

Reboiler is the first equilibrium stage, that is tray 0 in fig. 3.5. Equations are:

x0C5H12 = xC5H12 = 0.98

L1x1j − L0x0j − V

0y0j = 0

L1hl,1 − L0hl,0 − V0hv,0 +Qreb = 0

x0j φl,0j = y0j φ

v,0j∑

j

x0j = 1∑j

y0j = 1

with: j = 1..Nc

Condenser is tray N, fig. 3.6. Its equations are:

3.5 distillation column 27

Figure 3.6: Condenser, tray Ns

xNsC4H10 = xC4H10 = 0.96

VNs−1yNs−1j − (LNs +D)xNsj = 0

VNs−1hv,Ns−1 − (LNs +D)hl,Ns −Qcon = 0

xNsj φl,Nsj = yNsj φv,Nsj∑j

xNsj = 1∑j

yNsj = 1

where: j = 1..Nc

For improving the reliability of the model it was used an efficiency value. Sincethe debutanizer column is a well known literature case, in many papers is reportedits actual Murphree efficiency, that is exploited in this model:

ηM =yij − y

i−1j

yij,eq − yi−1j

This efficiency is referred to the vapor phase. The value used is 0.9 (Committee,1958) and it is an approximation, since in reality this value will vary from tray totray (and more from rectifying section to stripping section) and only 4 species areconsidered, instead of all the possible in reality. Equilibrium equations of all thestages but reboiler and condenser have to be re-written:

yijφv,ij = ηMx

ijφl,ij + yi−1j φv,ij (1− ηM)

with: j = 1..Nc

28 case study

Finally, it is worth remembering that the number of stages and the feed stagehave to be changed from the main code, so they must input variables. This foresightis mandatory because the optimization algorithm will have to find the best num-ber of theoretical stages and the feed tray, according to a cost function to minimize.

3.6 solver

The solver used for all the non linear systems presented in the code is the BzzNon-LinearSystem (Buzzi-Ferraris and Manenti, 2014). This is a class that is designedto exploit mainly the quasi-Newton algorithm to reach the solution. The issue isthat the BzzNonLinearSystem is a general solver, so it does not take advantage ofthe peculiarities of distillation problem.

First of all a Newton-Raphson based solver or a Inside-out method would havebeen more appropriate. Secondly many derivatives can be calculated analyticallyand many others can be simplified if the dependency is really small (Kister, 1992).All the remain derivatives can be evaluated numerically but the perturbation hasto be conform. A too small perturbed value will lead the derivatives to be meaning-less, whereas a too big perturbed value will move the solver far from the solution.But it is still possible that the problem is still too hard to solve, in this case othermethods have to be coupled and a relaxed problem can be formulated and solvedfor obtain good first tentative values.

Nevertheless, writing the code many expedients were taken in account in or-der to reach the solution at each call of the class BzzNonLinearSystem. An ad-hocsolver is not been developed for the time it could require. It is worth to mention acouple of these expedients:

• All the equations are written in order to have the unknowns in the maindiagonal of the Jacobian. This was done for the purity specifications too.1

• Partial molar balances are divided by the inlet molar flow of the column andthe enthalphy balances are divided by the enthalphy of vaporization of thefeed. In this way all the system is better conditioned.

• First guess values are weighted according to the inlet composition.

Moreover, a first attempt was made in Matlab® computing environment with itsfsolve solver, but the convergence of this linear system was requiring times of ∼ 100[s]. This time was reduced in C++ to less than 1 second. The reason of the highertime required by Matlab deals with its high-level syntax. Calling routines is verytime-consuming for Matlab and often these routines are implemented for solving

1 Note that changing the number of plates the system will increase or decrease.

3.7 diameter evaluation 29

Figure 3.7: Sparsity pattern of the Jacobian matrix. Partial condenser, 21 trays

general problems, not specific ones. For example the routine roots for finding thesolutions of Eos SRK each time an equilibrium problem is facing; this routine forsolving analytically a cubic equations had to be implemented in C++. The attemptof giving the jacobian to the Matlab fsolve was done and the time was reduced ofone order of magnitude, but it was still too long.

As an example, the Jacobian matrix of the first program in Matlab is reportedin fig. 3.7. There is not a total condenser but a partial one, since it was a first trialand in this condition the convergence of the solver is easier. The formulation hereis still not the best one, but almost all the unknown are in the main diagonal, spec-ifications too, both at the top and at the bottom of the column. The longer lines arethe enthalpy balances, shortest ones are about specification and stoichiometry. Asit is possible to notice there are 21 trays and nz is the number of points (deriva-tives) in the plot.

3.7 diameter evaluation

To estimate with accuracy the cost of the column, the diameter is needed. This onedepends on the volumetric flow-rate and from the property of the fluid; thanks tothe thermodynamic model it is possible to obtain from the compressibility factoralmost all the data required for each different stage. Hence the guidelines of Kister,1992 are followed. Some values are hypothesized in order to avoid a trial and errorprocedure. Values are summarized in table 3.5.

30 case study

parameter symbol value units

Surface tension σ 10 dyne/cm

Hole diameter dh 12.5 mm

Spacing S 24 in

Clear liquid height hct 2.5 in

Table 3.5: Design Hypothesis

Total area of the column’s section (AT ) is given by the summation of the net area(ANet) and the down-comer area(ADC):

AT = ANet +ADC

The first step is to determine the net area. This is done assuming, as a safetymargin from flood, that the column is designed for 80% of flood. The Souders andBrown flooding constant, corrected by Kister and Haas, is:

CSB = 0.0277(d2hσ

ρL

)0.125(ρGρL

)0.1(S

hCT

)0.5

This value has to be calculated for the bottleneck trays in both stripping andrectification sections. These two trays are the ones for which the C-factor (CS) ishigher.

CS,i =ViANet

√ρV,i

ρL,i − ρV,iwhere: i = 1..Nc

Where ANet is the net area, still unknown. The minimum upward vapor velocitythat would cause flooding (uS,flood) is then calculate as:

uS,flood = CSB

√ρL − ρVρV

Assuming a System Factor (SF2) of 0.9 (standard value for a debutanizer columnaccording to Gorak and Sorensen, 2014) it is possible to calculate the net area of

2 System Factor is an experimental derating number that keep in account foaming tendency too andit is used to reduce vapor and liquid velocities.

3.7 diameter evaluation 31

the both sections of the columns.

AN =V

0.8 uS,flood SF

The downcomer area is then sized in order to have a liquid velocity in the mid-dle of the recommended range for low foaming tendency with a 24-in spacing:

ADC =L

uDC; where: 0.1524 < uDC

[ms

]< 0.1829

The total area for both the sections of the column can now be obtained as sumof the net area and the downcomer area. If the difference between the two areasis lower than a 20%, it is likely not economically advantageous to consider twodifferent diameters, so only the biggest would be taken in account.

DT =

√4ATπ

In the column analyzed the difference between the 2 diameters is always lowerthan a 20%, so only the greater diameter is taken in account. This one will befundamental for the cost analysis and the net area will be used for the hydraulicsconstrains (flooding evaluation).

4O P T I M A L D E S I G N

In order to find the optimal design parameters of the distillation column -these arenumber of stages and position of the feed-, a function including both capital costs(CAPEX) and operative costs (OPEX) is minimized.1 In this section is explainedhow this minimization problem has been set and solved.

4.1 objective function

Concerning the evaluation of the separation’s cost, a Total Annual Cost function(TAC [$]) to be minimized was chosen from Kiss, 2013a.

TAC [$] = OPEX +CAPEX

payback period

A payback period of 3 years and 7920 h/year operating time (11 months a year)was assumed. The utilities are cooling water and LP steam. Prices are typical fora US plant. The total investment costs (CAPEX) includes the distillation columnwith sieve trays and the two heat exchangers.

CAPEX [$] = Column cost+Sieve trays cost+Boiler HE cost+Condenser HE cost

The standard cost correlations used are reported below:

Column cost [$] =M&S280

975.9D1.066H0.802(2.18+ FC)

1 On the contrary the diameter is chosen according to the procedure previously explained, hence theoptimal one for the nominal conditions and for the specified number of trays and feed location.

34 optimal design

Where M&S is the Marshall & Swift equipment cost index (M&S=1593.7 in 2017),D is the diameter of the column [m], and H its height. FC factor keeps in ac-count materials used and operative pressure (P [bar]). Since Carbon Steel is used(Fm = 1), only the pressure factor (Fp) is taken in consideration:

FC = FmFp = Fp

Fp = 1+ 0.0074(P− 3.48) + 0.00023(P− 3.48)2

H [m] = SNTray

Where S is the distance between plates (Spacing, [m]), assumed equal to 0.6 [m]because is the standard and the minimum that allows easy cleaning of the inter-nals.

Sieve trays cost [$] =M&S280

97.2D1.55H Fm

Heat Exchanger cost [$] =M&S280

474.7 A0.65(2.29+ Fd)

A kettle type reboiler (Fd = 1.35) and a floating head condenser (Fd = 1) werechosen. The area is simply evaluated from the duty to be supplied:

A[m2]=

Q

U LMTD

Where the heat exchangers were assimilated to double pipe controcurrent ones.Th simplification is strong, but allows to avoid to model them. Overall heat-transfercoefficients are taken from Perry and Green, 2007 and reported below in table 4.1;in the last line there are the ones used. It is mandatory to remind that these val-ues are a rough estimation of the real ones. Units are in

[Btu

◦F·h·ft2].2 Moreover, to

2 Remind: 5.678[

Btu◦F · h · ft2

]= 1

[J

K · s ·m2

]

4.2 algorithm 35

simplify further the model, the LMTDs (logarithmic average of the temperaturedifference between the hot and cold feeds at each end of the double pipe heat ex-changer) have been fixed (table 4.2). This approximation is not feasible, but it is aquick way to avoid the problem and focus more on the flexibility study.

reboiler : condenser :fluid steam-heated water-cooled

Propane 160 95

Butane 155 90

Used 150 90

Table 4.1: Overall Heat-Transfer Coefficients in Refinery Services

Operative costs (OPEX) are mainly the utilities’ costs, estimated from the duties(here in [GJ/s]) and the Operative Time (again 11 months, expressed in [s]). Costof other utilities, such as pumps or compressors, are not taken in account, due totheir lower impact to the economy of the process and for further simplification ofthe model.

Cwater,cold [$] = 4.43QconOT

Csteam,LP [$] = 7.78QrebOT

LMTD cost

utilities k $ ·GJ−1

Cold water 25 4.43

Low-Pressure steam 30 7.78

Table 4.2: Utilities costs

4.2 algorithm

The optimization of a single column is a typical mixed integer non-linear prob-lem (MINLP) and multiple locally optimal solutions are possible. Nevertheless,this case study has been chosen for its simplicity and the global minimum canbe easily guarantee, because the number of possible configurations is limited. Aconventional procedure (Patrascu and Bîldea, 2017), based on two different actions,

36 optimal design

was exploited to write the algorithm and minimize the TAC function.

• By sensitivity analysis, determining the feed tray that gives the minimumreboiler duty.

• Changing the number of trays and repeating the previous step until the theminimum of TAC is reached.

A column with a high number of plates (50) and the feed tray in the middle (25)is the starting point of the algorithm. Results in C++ are exported in a ASCII fileand then imported and plotted with Matlab®, since it is a very convenient tool fordata visualization and analysis.

Before illustrating the optimal design of this column, a note must be done. Sincethis problem is a typical MINLP, many attempts were done to implement it ina high-level modeling system for mathematical programming and optimizationsuch as GAMS®, in order to exploit highly efficient MINLP solvers as CONOPT orSNOPT. If the simplified solution, the one based on thermodynamic models suchas perfect gases and ideal liquid, was reached with not a big programming effort,on the contrary it was not possible to implement an SRK EoS. In fact none -tothe author knowledge- has ever implemented a cubic Eos directly in GAMS, butalways indirectly, coding an Dynamic-Link Library (such it was done in Kossack,Kreamer, and Marquardt, 2006 for NRTL coefficients). Many tries have been doneto develop an ad hoc SRK Dynamic-Link Library for GAMS, but unsuccessfully,since not enough documentation was found about it.

4.3 result, comparison with a simulation software and discussion

The converge of the model to the minimum of TAC=169970.0 [$/yr] is showed inFig. 4.1. The result is a column with 22 stages (counting only equilibrium ones,hence the reboiler but not the condenser) and the feed in the 12 tray (if reboiler is0). All the specifications are respected. A comparison of the internal profiles withthe commercial software Aspen HYSYS® with the same optimal set-up is showedfrom fig. 4.2; in table 4.3 are summarized the most interesting results.

There is a generally good agreement, with only slightly differences, maybe mainlydue to the different thermodynamic parameters used in equations. Temperatureprofile inside the column suggests a slightly different temperature in the feed tray:maybe it would have been better to position the feed tray one stage above accord-ing to the simulations done in Aspen. Areas may seem relatively small, but this isdue both to the small feed entering in the column and the strong approximationsin designing the heat exchanger (LMTDs were fixed).

Concerning the resulting diameter, this suits perfectly the hydraulic plots in As-pen HYSYS®; an annotation arises because for a such small diameter is usually

4.3 result, comparison with a simulation software and discussion 37

c++ aspen unit

Number of stages 22 22

Feed stage 12 12

Condenser duty -1.0634E6. -1.080E6 kJ/h

Reboiler duty 6.9896E5 7.177E5 kJ/h

Reflux Ratio 1.08 1.11

Diameter 0.57 m

Condenser’s Area 23.14 m2

Reboiler’s Area 7.6 m2

Diameter 0.57 m

OPEX 80378.4 $/yr

CAPEX/3 89591.6 $/yr

TAC 169970.0 $/yr

Table 4.3: Results and comparison with Aspen Hysis

Figure 4.1: Minimization of TAC [$/yr]

38 optimal design

Figure 4.2: Temperature Profile inside the column

Figure 4.3: Molar flow rate profiles inside the column

4.3 result, comparison with a simulation software and discussion 39

Figure 4.4: Molar fraction profiles inside the column

Figure 4.5: Molar fraction profiles inside the column

40 optimal design

suggested to use packings instead of sieve trays. But since the object of this thesisis the methodology and since the threshold value in Aspen for the diameter is 0.6[m] (that is really close to the result obtained), no further changes will be done tothe case study of Hoch, Eliceche, and Grossmann, 1995.

Moreover it is clearly that the hypothesis of a spacing of 24 inches is incorrect,since it is too much for a tower with a 0.57 meters of diameter. Such a big valuewas chosen because it is the smallest that allows an easy cleaning of the column’sinternal; indeed probably 80% of the world’s trayed columns employ spacing of 24inches (this has been the industry standard-forever). A lower value like 18 inchesshould be used instead, but this will reduce the turndown ratio of 2:1. By the waythis was not done in this thesis since developing and commenting the approach isthe main goal.

Finally, as further comparison and curiosity, the results from the FUG methodare reported in table 4.4. Specification used are a n-butane 0.02 light key molarfraction in the bottom distillate and a 0.240 n-pentane heavy key molar fractionin the top distillate (in order to have a 0.96 molar fraction of n-butane in the topdistillate). It has to be notice that here no efficiency is applied.

variable numerical result rounded

Minimum number of stages 7.75

Minimum reflux ratio 0.937

Minimum reflux ratio * 1.2 1.124

Number of stages 19.16 20

Position of feed 7.8 8

Table 4.4: Results of FUG method. No efficiency is taken in account

Results confirms quite well the previous optimization. It is expected a slightlylower number of trays, 20 instead of 22, but it is because of the efficiency of 0.9 wasnot considered in FUG method. Due to the close ideality of the separation, it seemsunnecessary to use a more complex model, since a rough one gives already a verygood result. Nevertheless, a rigorous model with MESH equations will be used inthe next sections. The reason lies behind the interest of knowing in detail duties(for evaluating OPEX) and internal profiles such as volumetric flowrates (for thehydraulics).

5F E A S I B I L I T Y C O N S T R A I N S

Since operational parameters and design uncertainties will vary in the followingChapters, it is necessary to verify if each solution of the column is feasible. Indetail, feasibility of the solution will highly dependent from the technology usedand its detailed description should be mandatory to have a reliable result. Nev-ertheless this work is focus on the methodology, so many approximations will becarried out. Concerning the hydraulics part, feasibility is verified if the floodingvelocity is greater than the real vapor velocity and if the weeping fraction is negli-gible; concerning the heat supplied and removed, a maximum absolute values forduties from the reboiler and to the condenser is fixed. The procedures below aretaken from Kister, 1992 and Perry and Green, 2007. In addition to this, it is worthto remember that specifications, defined previously as molar fraction purity, areequality constraints and they have to be always satisfied.

5.1 flooding constrain

Flooding is an excessive accumulation of liquid inside a column. It can cause arapid increment in pressure drop, reduction in bottom flow-rate, liquid exitingfrom the top of the column and instability in general. Flooding is usually theupper capacity limit of a distillation column and its estimate is used to size thediameter of the column. This was done in the previous section, where the towerwas designed in order to operate at 80% of flood.

Initially, using the C-factor correlation, the bottleneck tray has to be found. Takenthe properties of the vapor and liquid phase in that tray it is possible estimate theminimum upward vapor velocity that can cause flooding (uC,flood).

umax < uC,flood

Since the actual Net Area (AoptN ) is known from section 4, it is possible to esti-mate the velocity in the critical tray and comparing this velocity with the minimumupward vapor velocity that will cause flooding.

42 feasibility constrains

VC

AoptN SF

< uC,flood

A check is done to ensure validity of this formulation. As it is noticeable from fig.5.1 the molar flowrate is linearly increased beyond the feasibility limit. Of coursethe flooding velocity will vary if compositions, pressure or other variables willchange, but in this plot is fixed. It is important to notice this because in the nextsteps other variable will change and so flooding velocity will be different for eachcase. In this simulation stage 20 has always the maximum C-factor, therefore herethe flooding check was automatically done.

Figure 5.1: Comparison between vapor velcity in the critical tray and flooding velocity atincreasing flow rate. All the other variables are fixed

5.2 weeping constrains

Weeping is liquid descending through the tray perforations, short-circuiting thecontact zone, which lowers tray efficiency. At the tray floor, liquid static head thatacts to push liquid down the perforations is counteracted by the gas pressure dropthat acts to hold the liquid on the tray. When the static head overcomes the gaspressure drop, weeping occurs.

Some weeping usually takes place under all conditions in an non uniform waydue to sloshing and oscillation of the tray liquid, but generally this weeping is

5.2 weeping constrains 43

too small to appreciably affect tray efficiency. The weep point is the gas velocityat which weeping first become noticeable: here some efficiency is lost and as thevelocity decreases, the lost will be more important. The operating limit is reachedwhen the weep rate is large enough to significantly reduce tray efficiency.

The main factor that affects weeping is the fractional hole area. The larger it is,the smaller will be the gas pressure drop and the greater the weeping tendency. Arigorous prediction of the weep point could be made according to the Lockett andBanik correlation since the column works at low pressure (<1100 kPa). Only thetray with the lowest CSB can be taken in account. All the passages can be found inPerry and Green, 2007 and in Chuang and Nandakumar, 2000.

Nevertheless, this method was not implemented for all the iterations since nota significant weeping fraction was encountered for the variations made. Indeedthis column is made of sieve trays and they have a usual turn-down ratio of 2:1.It means that the minimum load is the half of the maximum load and actuallyin the code such a small value of load is never reached. A higher turn-down canbe obtained with a different technology, such as valve trays, but the cost wouldbe bigger and the difficulty in cleaning the tower would grow. Details of differenttrays are reported in table 5.1 (Gorak and Sorensen, 2014).

deck type cost turndown

Sieve Low 50%

Fixed valve Medium 40%

Moving valve High 30%

Bubble cap Very high 10%

Table 5.1: Comparison of deck trays

In figure 5.2 it is reported a more detailed, but still approximated, descriptionof the tray hydraulics. The operation of a tray should stay inside the area, wherea good efficiency is ensure. Increasing the loading, the point will move up alonga straight diagonal with a positive coefficient, until a flooding region, where thefroth totally fills the tray spacing; whereas decreasing the loading will move downthis line, until weeping will start: small weeping fractions are tolerable (thresholdvalue was 3% in the past, now it is thought to be about 20%), but at higher valueit affects efficiency. At the dump point the tray starts to work as a dual flow tray.The dependency flooding vs tray efficiency can be highlighted in figure 5.3: whilethe point is inside the area (of figure 5.2), differences of efficiency exist, but theyare small. As soon as the point exit from this region, efficiency will drop drownand operations (hence purities) are no more assured.

44 feasibility constrains

Figure 5.2: Performance diagram: simplified hydraulics of a tray. Area of satisfactory op-eration is shaded

Figure 5.3: Tray efficiencies across a broad range of loadings

5.3 duty constrains 45

5.3 duty constrains

Both reboiler and condenser can supply and remove a limited heat. An assump-tion has been made: the maximum absolute value is the nominal value plus a 30%(table 5.2). This assumption simplify a lot the process because does not require tomodel both the heat exchangers. A more detail model can be implemented but itwas not done since the work is focused more on the flexibility study.

nominal upper boundary unit

Condenser duty -1.0634 E6. -1.38242 E6 kJ/h

Reboiler duty 6.9896 E5 9.08648 E5 kJ/h

Table 5.2: Nominal and limit values for duties

6F L E X I B I L I T Y I N D E X

In this part it is assigned a Flexibility Index to the debutanizer column and thegrowth in flexibility is related with OPEX. A similar work was already done byHoch, Eliceche, and Grossmann, 1995, but results are not comparable since all thedata used (properties of compounds, constants and details of the model) were notavailable.

6.1 formulation

An arbitrary set of uncertain parameters of different nature is chosen, in order toillustrate an example of the possible applications. These are the three inlet molarflow rates, the Murphry efficiency and the Pressure inside the column. All of them,with their maximum and minimum expected variations, are reported in table 6.1.It is important to emphasize that the choice of the set of uncertainties and theirexpected variation is a task that requires a high knowledge of the process and itsweakness. Therefore the values used below are just an example to show the capa-bilities of this method.

θk θNk uncertainty ∆θ+k ∆θ−k unit

FC3H6 0.2 50% +0.1 -0.1 kmol/hr

FC3H8 0.2 50% +0.1 -0.1 kmol/hr

FC4H10 24 30% +7.2 -7.2 kmol/hr

FC5H12 12 30% +3.6 -3.6 kmol/hr

P 4 25% +1 -1 bar

ηM 0.9 3% +0.027 -0.027

Table 6.1: Uncertainties considered. Nominal values and maximum expected deviations

The nominal value θNk of each uncertainty θk is in the second column of thetable, whereas on the right there are the respective maximum and minimum ex-pected variation ∆θ+k and ∆θ−k . In this example they are assumed to be equal, butthis is not always true. It has to be noticed that since feed’s composition changes,also its initial temperature will vary, because the inlet flowrate is at bubble temper-ature; therefore also the feed temperature has to be avaluated for each case. The

48 flexibility index

Flexibility Index problem anticipated in Chapter 2 is reported here:

FI = max δ (6.1)

s.t. maxθ∈T(δ)

minzmaxj∈J

fj(d, z, θ) 6 0

T (δ) ={θ |(θN − δ∆θ−

)6 θ 6

(θN + δ∆θ+

)}

The solution is the maximum δ that makes the column still feasible. Expecteddeviations ∆θ+k and ∆θ−k correspond to a Flexibility Index equal to 1. Since thereare 6 uncertainties, the vertices of the hyper-rectangle is going to be analyzed are26 = 64. The algorithm will increase the value of δ of a small quantity at each itera-tion, until the first bottleneck is reached (a constrain is violated). Control variablesz are reflux ratio and product flowrates.

This case study is a conventional distillation column with a quasi ideal liquid-vapor equilibrium, so it is known a priori which are the worst conditions, hencecombinations of uncertainties θk, that would take the column to an unfeasiblepoint. Nevertheless this case study was used to show a systematic methodologythat could be applied in the future to more complex case study, such as sequencesof distillation columns with complex mixtures.

6.2 results

In figure 6.2 it is possible to see all the simulations done. Each point requires afraction of second of cpu-time on a Intel i5-7200 CPU laptop with 8 GB of RAM. Itappears clearly that there is a linear dependence for each vertex between flexibilityand OPEX. This is because heat exchangers are not modeled but OPEX are directlyproportional to duties required and because variations are small.

The Flexibility Index of this column is found to be 0.57 because at this value of δthe first bottleneck is met (table 6.2). The reason is the flooding constrain violated,because vapor velocity is higher than the maximum allowable for the designedNet Area. The conditions at which this unfeasible point occurs are low pressure,high flow rates and low efficiency. It is worth to notice that the OPEX at this ver-tex are not maximum as one can think. Hence for increasing the Flexibility of thiscolumn the first action would be increasing the minimum pressure to avoid highflows or as alternative increasing the diameter of the column, that is not possibleif the tower is already built. It is possible also to remove the downcomer (dual-flow trays), but if this action can usually improve the capability of a column andmake the handling of fouling easier, for sure it slightly reduces its efficiency andincreases a lot the instability. Dual-tray flow is hence a valid revamping alternative,since possible channeling is avoided (the diameter of the column is lower than 8

6.2 results 49

flexibility opex [$] FC3H6 FC3H8 FC4H10 FC5H12 P ηM

0.57 91819.3 0.257 0.257 28.104 14.052 3.43 0.88461

0.58 92009.9 0.258 0.258 28.176 14.088 3.42 0.88434

0.59 92200.1 0.259 0.259 28.248 14.124 3.41 0.88407

0.59 91469.4 0.259 0.259 28.248 14.124 3.41 0.91593

0.60 91066.4 0.140 0.260 28.320 14.160 3.4 0.88380

0.60 91076.4 0.260 0.140 28.320 14.160 3.4 0.88380

0.60 92390.0 0.260 0.260 28.320 14.160 3.4 0.88380

0.60 91647.0 0.260 0.260 28.320 14.160 3.4 0.91620

Table 6.2: First combinations of critical uncertainties for which flooding constrain is vio-lated

ft), but better alternatives exist. If the column is an old one, a more complex re-vamping is possible. In figure 6.1 is reported a qualitative graph of two possiblealternatives of revamping an old column. Capacity can increase from 5% up to 20%if piping and ancillary equipments as heat exchangers, pumps, etc. can handle thisincrement. The simple one-to-one tray revamp is cheaper than a more complexthree-to-two revamp (i.e. 60 trays of 400 mm instead of 40 trays of 600 mm), butdoesn’t improve much the flexibility of the column; complex revamps are preferrednowadays.

The decision to explore further the uncertainties space is done. Now the situationbecome more interesting: as Flexibility increases, profiles start not to be so linearbut slightly exponential. This is due to the fact that the reflux ratio is increasing be-cause the physical limit of the separation is getting closer. It is noticeable that alsoall the points after the first bottleneck are not more feasible: actually those verticeshave gone out the feasibility region and cannot return inside anymore. Moreover,as it is possible to see, a second bottleneck at higher Flexibility Index is encoun-tered(at FI=0.74). Here the problem is that the duty subtracted to the fluid in thecondenser is not enough. A colder fluid or a bigger area of the heat exchangercould de-bottleneck the column, but the Flexibility Index would remain still 0.57,since the first restriction is the flooding. No weeping is observed for these smallvariations. 1

Finally in fig. 6.3 the profile of Flexibility is showed. Dependency is clearly lin-ear in the first part of the diagram and this is due to the proportionality betweenflow-rates and duties required at the same composition. Someone would have ex-pected a non linear dependence if OPEX had not been linearly dependent fromduties, hence if the reboiler and the condenser had dad been modeled rigorously.On the contrary in the second half of the diagram there are vertices’ profiles that

1 Actually there was the intention to avoid weeping because in the further chapters weeping wouldhave increased drastically the time of the optimization. As a consequence this set of uncertaintiesand these small variations were chosen.


Figure 6.1: Two possible alternatives of revamping an old column

Figure 6.2: Debutanizer column: Flexibility vs Opex. The Flexibility index is equal to 0.57.Light blue: flooding constrain violated. Red point: duty constrains violeted

6.3 remarks 51

flexibility opex [$] FC3H6 FC3H8 FC4H10 FC5H12 P ηM

0.74 103942.7 0.274 0.274 29.328 14.664 4.74 0.88002

0.75 104277.2 0.27500 0.275 29.400 14.7 4.75 0.87975

0.76 104612.0 0.27600 0.276 29.472 14.736 4.76 0.87948

0.77 104947.3 0.27700 0.277 29.544 14.772 4.77 0.87921

0.78 105283.1 0.27800 0.278 29.616 14.808 4.78 0.87894

0.79 105619.3 0.279 0.279 29.688 14.844 4.79 0.87867

0.79 104057.9 0.279 0.2790 29.688 14.844 4.79 0.92133

0.80 105956.0 0.280 0.280 29.760 14.880 4.8 0.87840

Table 6.3: First combinations of critical uncertainties for which duty constrains is violated

are clearly non linear. The reason is the difficulty of the separation that is increas-ing when n-butane and n-pentane have a similar molar flow rate (it is noticeableto remember that specifications about molar fractions are fixed). In this case dutiesand reflux ratio have to increase, hence an higher cost has to be payed.

6.3 remarks

This method has showed to be easily applicable once the model of the column isbuilt and it could be a very simple tool for comparing different configurations dur-ing the design phase and when a revamping is needed. Moreover, it was noticeablethat, if deviations are small, it could be not necessary to analyze all the vertices,since from the designer’s knowledge or from a initial sensitivity analysis, it is pos-sible to determine the critical vertices and save cpu-time to study only their growthin the space of uncertainties. But if deviations are getting bigger, it is possible toexpect non linear behavior and then a complete analysis could be useful to reducethe uncertainties space in a 2-D map, flexibility vs OPEX.

Nevertheless the weak points are many. First of all the methodology is verysensitive to the initial decision: the decision maker has to select the main uncer-tain parameters and more in detail their expected deviation to give a scale for acomparison. The Flexibility Index will be strongly related to these decisions anddifferent configurations can not be compared on various scales, but only on thesame. Secondly, its biggest strength is also the weakest point: only vertices aretaken in account, all of them with the same importance, as they could be equallyprobable, and no considering all the possible linear combinations. It would havebeen more interesting relating uncertainties to profits, no to OPEX, and creatingiso-profit lines in the space of uncertainties, hence fixing them as variables andoptimizing according to a profit. This was the idea from that the second method-ology raised. Finally, if the Flexibility Index gives an insight on the capability ofthe process, hence the first bottleneck and the expected OPEX, this information


Figure 6.3: Flexibility vs OPEX, maximum expected costs in the range of feasibility

are still too roughly for deciding which configuration adopt, if processes are verysimilar.

7F L E X I B I L I T Y A N A LY S I S

The second approach will be applied to the debutanizer column in this Chapter.The aim is to maximize the range of operation of the process with the minimumvariation of the average profit. Since there is a competition between these two dif-ferent targets, goal programming is used to find a compromise between flexibilityand profits.

7.1 goal programming

The optimal design was already found in section 4, so from that chapter sizes of thecolumn are taken. On the contrary, uncertainties, in which the process is wantedto be optimized in, are reduced to inlet molar flow-rates; this is done to reduce thetime required to reach the minimum. Since species are 4, there will be 8 variablesin the optimization, a lower value and an upper value that determine the rangeof flexibility for that inlet molar flow rate. This variables are named θ−j and θ+j ,where j here is referred to the species (j = 1...4). Each variable must have an upperand a lower boundary (θU and θL) and a starting value (θg). It has been chosenas boundaries the expected variations used in the previous section (θN −∆θ− andθN−∆θ−), and as starting values the ones close to the nominal point N (table 7.1).

flow θ−j,L < θ−j,g < θ−j,U θ+j,L < θ+j,g < θ+j,U

FC3H6 0.1 0.2-ε 0.2 0.2 0.2+ε 0.3 kmol/hr

FC3H8 0.1 0.2-ε 0.2 0.2 0.2+ε 0.3 kmol/hr

FC4H10 16.8 24-ε 24 24 24+ε 31.2 kmol/hr

FC5H12 8.4 12-ε 12 12 12+ε 15.6 kmol/hr

Table 7.1: Starting values, upper and lower Boundaries.

If the column has showed to have a Flexibility Index of 0.57 in the previous sec-tion, now, since neither pressure of efficiency can variate, the Flexibility Index ishigher. Indeed value for tray efficiency and internal pressure are like at the begin-ning 0.9 is 4 bar. Constrains are again on flooding, weeping and duties. The pricesfor both the distillates, top and bottom, are calculated in order to nullify both Opexand Capex in the time window of three years.

54 flexibility analysis

Figure 7.1: Flexibility analysis problem in two dimensions. The first aspiration level is rep-resented by the blue rectangle, composed by the expected deviations, whereasthe second aspiration level is represented by an average profit of vertices (redpoints) as close as possible to the profit of the nominal value N of uncertainties

As it was introduced in Chapter 2, the problem is formulated as follow:

Min (F1, F2) =

=Min

ψ1(Πk


)− 1

)2+ψ2

∑w


Nw · ProfitN

2

with: w = 1...Nw,

k = 1...p


Where Nw is the number of w vertices (24 = 16), p the number of uncertaintiesk (here only the 4 flow-rates) and θ+k and θ−k the variables to optimize (8 in total,the lower and upper limit of each range considered in a single iteration).

In figure 7.1 it is possible to visualize the problem in two dimensions. The firstaspiration level is the volume of the hyper-rectangle composed by the expecteddeviations. But since variables have different orders of magnitude, it has been cho-sen to adimensionalize each side of the hyper-rectangle. For this reason the targetvolume goes to one. It is important to remind that in this case it is not known

7.2 algorithm 55

if the hyper-rectangle can reach the target, since feasibility constraints are active.The second aspiration level is the profit at the nominal value, so an average of theprofit in vertices has to be done. Physically this means that every vertex has thesame probability and, in this sense, profits in some vertices will compensate othersvertices in order to have an average value as close as possible to the nominal one.Values used for linear scalarization are ψ1 = 1 and ψ2 = 1.E2; for this values F1and F2 start to have a similar order of magnitude and their sum start to be a func-tion with a smooth profile. Higher values of ψ2 can be used and a Pareto diagramcan be build as a valid tool for the decision maker.

It was chosen to use profit instead of OPEX because it was thought more usefulreasoning about earnings instead of costs. Indeed the goal is to find the maximumregion in which the average profit is still close to the nominal one. If in the previousChapter operative costs were used for literature reasons and because uncertaintieswere considered unavoidable, now the object is to find this optimal hyper-rectanglein the space of uncertainties; in this way, during the operation, if an average feedgets out from this region, an action could be mandatory to correct a probable loss.

Prices were calculate from the nominal case, so that the products flow rate canzeros both CAPEX and OPEX, where all the CAPEX were divided in three yearsand OPEX in 11 months a year. Then, for each vertex at each iteration, profit is cal-culated as product between price and flowrate minus operative costs. In this wayprofit will always be a positive value. It is remarkable that the time base is alwaysthe same. Finally, it is important to notice that prices of bottom and top distillatesare assumed equal in molar base; the reason is because again the interest of thiswork is discussing the methodology.

7.2 algorithm

The solver used belongs to the class BzzMinimizationMultiVeryRobust of the BzzMathlibrary (Buzzi-Ferraris and Manenti, 2014). This robust optimizer is based on theparallel computing algorithm by Buzzi-Ferraris and Manenti, 2010, which is moreeffective to handle discontinuities and very steep creeks/narrow valleys problemstypical of chemical systems. It plays at two different levels: outer search for theglobal optimum and tackling possible narrow valleys basing on the same conceptof design of experiments techniques; inner search to quickly find a local minimumon the bottom of the valley or close to discontinuities.

Moreover, in order to avoid unfeasible vertices during the optimization process,the function Bzzunfeasible is called each time a hydraulic or duty constrain is vi-olated, in order to exclude that unfeasible vertex, and so all the iteration. Initialvalues for the variables are very close to the nominal points, as it is possible to seefrom table table 7.1.


7.3 results and discussion

The minimization process of the objective function is showed in fig. 7.2 as well asprofiles of concentration in fig. 7.3 and 7.4. The iteration with the lowest minimumreached is marked with a dotted line. For each iteration 16 column simulationshave to be done and all the constrains have to be checked. There is no way toprove that the global minimum is reached and rightly all the unfeasible iterationsare not reported in the plot. The whole problem took a couple of hours of cpu-timeon a Intel i5-7200 CPU laptop with 8 GB of RAM. However the RAM used is neg-ligible (few MB).

Figure 7.2: Feasible iterations. Flexibility goal is ψ1F1; profit goal is ψ2F2. Dotted line isfor indicating the minimum

At the starting point the values of variables are at nominal conditions, so, asit is possible to notice, ψ2F2 has a minimum (average profit equals to the nomi-nal one) and ψ1F1 a maximum with a value of one (no flexibility considered). Asthe number of iterations increases the two functions get closer, but generally theflexibility contribute is more difficult to accommodate: this happens because thefeasibility constrains do not allow the hyper-rectangle to expand as the optimizerwould, (following the opposite of the gradient into the unfeasible region). On thecontrary it seems easier to fix the profit in order to have an average one close tothe nominal value.

7.3 results and discussion 57

Figure 7.3: Upper and lower profile of propylene and propane. Dotted lines for minimumand boundaries, traced ones for starting values

It is worth to notice that the upper limit of the butane molar flow rate is notreached, since the constrains, mainly on flooding, are active. And since there isnot an high flowrate of butane, also the lower boundary is not reachable, becausethere will not be an upper value to compensate it in function ψ2 (Profit). The samebehavior is observed for pentane is ψ2 value is increased.

The limits of propane and propylene are the same, but the algorithm found thathaving a wider range of propylene and a tighter one of propane in the system ischeaper. This is more noticeable if the order of magnitude of ψ2 is increased.

Results are in table table 7.2 and table 7.3.

flow θ−j,L < θ−j,Opt < θ−j,U θ+j,L < θ+j,Opt < θ+j,U

FC3H6 0.1 0.1 0.2 0.2 0.3 0.3 kmol/hr

FC3H8 0.1 0.1 0.2 0.2 0.3 0.3 kmol/hr

FC4H10 16.8 18.247 24 24 28.917 31.2 kmol/hr

FC5H12 8.4 8.4 12 12 15.6 15.6 kmol/hr

Table 7.2: Results of the Flexibility analysis for ψ1 = 1, ψ2 = 1E2. Molar flow rates


Figure 7.4: Upper and lower profile of butane and pentane. Dotted lines for minimumand boundaries, traced ones for starting values

function value goal

ψ1F1 0.06706 Flexibility

ψ2F2 0.00990 Profit

Sum 0.07696

Table 7.3: Results of the Flexibility analysis for ψ1 = 1, ψ2 = 1E2. Functions’ values

7.4 additional optimization

A second optimization is done increasing further the value of ψ2, from 1.E2 to1.E4. Doing so the importance of gaining an average profit as close as possible tothe nominal one will grow. Results are in table 7.4 and table 7.5, more than 6000

iterations were done.

As it is possible to notice there are not such as a big differences in result, com-paring this optimization (ψ2 = 1.E4) to the previous one (ψ2 = 1.E2), but the lowervalue of n-butane and n-pentane molar flow rate. In particular the first value makesthe n-butane range a bit wider, whereas the second causes a very narrower range ofn-pentane. This difference lets understood that the minimum is different: indeed inthe previous optimization more importance was given to flexibility, whereas nowit is fundamental that the profit is closer to the nominal one.

7.5 remarks 59


FC3H6 0.1 0.10143 0.2 0.2 0.29976 0.3 kmol/hr

FC3H8 0.1 0.10137 0.2 0.2 0.3 0.3 kmol/hr

FC4H10 16.8 17.92997 24 24 28.91762 31.2 kmol/hr

FC5H12 8.4 10.01868 12 12 15.59990 15.6 kmol/hr

Table 7.4: Results of the Flexibility analysis for ψ1 = 1., ψ2 = 1.E4. Molar flow rates

function value goal


ψ2F2 0.0002 Profit

Sum 0.17448

Table 7.5: Results of the Flexibility Analysis for ψ1 = 1., ψ2 = 1.E4. Values of functions

7.5 remarks

Even if it is not possible to assign a simple scalar to directly correlate the processto the flexibility, as it was done in the previous Chapter (6), now more detailedinformation about the process can be collected. With this new procedure criticalvertices are found and they are optimized according to profit. The result is a set ofranges for operability, one for uncertainties. Therefore if using a Flexibility indexis useful to compare different configuration, here this new flexibility analysis caninvestigate more in detail the uncertainties space, exploiting the different values ofscalars psii too.

Nevertheless, it is important to remind the physical meaning behind the averag-ing of profits: every vertex will have the same probability, so it is like designinga process considering only vertices and none of the internal points, neither thenominal point. A greater value of profit will compensate lower values. It could bepossible to change the weight of each vertex according to the knowledge of the pro-cess, in order to give more importance to the more probable; but the best approachwould be to find a new way to estimate more rigorously the average profit of thearea, since it is obvious that using only vertices is a very approximate choice. Asit was mention in Chapter 2, the best way would have been to discretize the entirehyper-rectangle of uncertainties with a grid and evaluate profits in every nods (orin the middle of each cell). At this point a different weight can be given to theprofit of each point according to the probability of the point to happen.

But this is very time consuming: if one simulation takes about 0.5 seconds, anoptimization with 4 uncertainties and a grid with a side of 10 points would need104 · 0.5 = 5000 seconds at each iteration, too much. Nevertheless, a convergenceanalysis can still be done on the solution to see how big is the error of consideringonly vertices and which could be the minimum number of nods to consider foreach side to reduce the error. Moreover, it must be noted that an equal number of


point for each side is not the proper choice, because considering an higher numberof points only for variables that modify more the profit (hence n-butane and n-pentane flow rates), it is possible to reduce the number of grid points. This aspectwill be analyzed further in Chapter 8.

8F L E X I B L E D E S I G N

Finally a new design method is presented, that does not take in account onlyCapital and Operative costs, but also Flexibility.

8.1 flexibility contribution

The idea is to exploit the last function for designing a new column, hence not onlyfor evaluating its flexibility. The design parameter that was chosen is the numberof trays. At each iteration the best feed tray is calculated minimizing the reboilerduty for the nominal condition and, according to the results of this inner optimiza-tion, the best diameter and the heat exchanger are evaluated as already explainedin Chapters 4. These values are not only necessary to estimate the new CAPEXof the column and to compare it with the starting optimal one, but they will in-directly influence OPEX too. Moreover, all the constrains have to be re-calculated,according to the new column, in order to check if all the vertices in the space ofuncertainties of that iteration are feasible. Hence from the Net area will be possi-ble to calculate the maximum allowable vapor velocity, from a detailed design theweeping fraction and from the nominal values of duties the maximum capabilities.The optimization problem was presented in Chapter 2 and it is reported below:

Min (TAC, F1) =

=Min

ψ3 CAPEXpayback period

+ψ4

∑wOPEXw

Nw+ψ5

(Πk


)− 1

)2with: w = 1...Nw,

k = 1...p


It is fundamental to notice that the trays’ number is an integer variable, so its be-havior will be very different from the other continuous variables. Nevertheless, thesame previous optimizer, BzzMinimizationMultiVeryRobust, is used, with the at-tention of rounding floating values to integer. This approach could not be efficientat all in finding the minimum, since the optimizer will try to evaluate numerically

62 flexible design


FC3H6 0.1 0.2-ε 0.2 0.2 0.2+ε 0.3 kmol/hr

FC3H8 0.1 0.2-ε 0.2 0.2 0.2+ε 0.3 kmol/hr

FC4H10 16.8 24-ε 24 24 24+ε 31.2 kmol/hr

FC5H12 8.4 12-ε 12 12 12+ε 15.6 kmol/hr

l 0 u

NTrays 19 22 29

Table 8.1: Variables of the optimization. Upper and lower boundaries and starting values

first and second derivatives of the function value with respect to the number ofplates, information that are misleading, since in reality the number of plate foreach tentative is always the same. Moreover, the optimizer will try to find a mini-mum, treating it as a continuous variable, but in reality it is using the same integervalue (hence wasting cpu-time in useless iterations). A more rigorous approach re-quires a MINLP to be used instead, or at least an algorithm that tries every singlediscrete integer values.

The uncertainties with their upper, lower and starting values are the same ofChapter 7 and are reported in table 8.1, together with the new variable, the num-ber of plates:

Values for Linear Scalarization are in table 8.2 and have been calculated with thesame approach explained in previous Chapters. Importance of Flexibility can byincreased further by increasing ψ5.

parameter value

ψ3 1

ψ4 1

ψ5 1. E5

Table 8.2: Parameters for Linear Scalarization

8.2 results and discussion

Profiles during the iterations are in figure from 8.1 to 8.4. The whole problem tooka couple of hours of cpu-time on a Intel i5-7200 CPU laptop with 8 GB of RAM.However the RAM usage is negligible (few MB). Again all the non feasible itera-tions (here mainly flooding constrain not respected) are not reported in plots.

It is very interesting to notice what was presented in Chapter 1 and was called"smart design", that is: a design that consider more theoretical plates and a lower

8.3 approximation of vertices 63

reflux, so lower flowrates and a smaller diameter. Indeed for this debutanizer col-umn the minimum of the function is found at 23 trays, hence one more than stan-dard rigorous optimization that consider only minimization of Total Annual Costs(TAC). Including a flexibility contribute this optimization leads the designer to con-sider a slightly more expensive column in favor of handling a wider range of feedwith a lower operative costs. A comparison between the old optimal design and thenew more flexible is shown in table 8.5. Looking at the results it is noticeable thatvalues are not so close to reality since functions and parameters used are genericand they have to be defined more in detail. For example areas and diameter havedecreased since they were optimized according to the number of plates, thereforeif this one was increased, they must decreased. Then again all the results must beconsidered only as a starting point for a more detailed design.

During the optimization only the flooding constrain is violated, hence againflooding should be the first and main bottleneck of the column. As a consequenceneither n-butane or n-pentane flow rates reach the upper values. But it is importantto emphasize that these values are not comparable with the results of Chapter 7,since the Multi Objective Function is different, i.e. here costs are taken in accountinstead of profits. Other constrains are not active, since deviations from nominalvalues are too small. It is important to remind that these constrains were activein Chapter 6 because other two variables were considered: pressure of the columnand Murphree Efficiency. Here they are fixed.

Contrary as it was expected, the BzzMinimizationMultiVeryRobust succeededin handling the integer variable number of trays. It is true that the profile is flat formost of the iterations, but the reason is because the minimum is there; indeed areexplored number of plates different to the best one, but there is not a minimum forthem. Nevertheless, there is the need of changing solver if one wants to implementmore integer values, as the feed’s tray or, in an extractive distillation, the solvent’stray.


FC3H6 0.1 0.1 0.2 0.2 0.3 0.3 kmol/hr

FC3H8 0.1 0.1 0.2 0.2 0.3 0.3 kmol/hr

FC4H10 16.8 16.8 24 24 29.82 31.2 kmol/hr

FC5H12 8.4 8.4 12 12 14.81 15.6 kmol/hr

Table 8.3: Results of the Flexibility analysis. Molar flow rates

8.3 approximation of vertices

As anticipated in Section 7.5, analyzing the process considering only vertices phys-ically means that the column will operate only in these critical conditions, hencenone of the internal points, neither the nominal one. In reality this has no sense,

64 flexible design

Figure 8.1: Values of contributes and their sum during the optimization

Figure 8.2: Upper and lower profile of propylene and propane during the optimization


Figure 8.3: Upper and lower profile of n-butane and n-pentane during the optimization

Figure 8.4: Number of trays during the optimization

66 flexible design

function value object

ψ3F3 90380.54 CAPEX

ψ4F4 77314.99 OPEX


Sum 187125.47

NTrays 23

Table 8.4: Function values at minimum

old design flexible design unit

Number of stages 22 23 [counting reboiler]

Feed stage 12 13 [counting reboiler]

Condenser duty -1.0634E6. -1.055E6 kJ/hr

Reboiler duty 6.9896E5 6.9097E6 kJ/hr

Reflux Ratio 1.08 1.06

Diameter 0.57 0.568 m

Condenser’s Area 23.14 22.965 m2

Reboiler’s Area 7.60 7.51 m2

Diameter 0.57 0.568 m

OPEX 80378.4 79606.5 $/yr

CAPEX/3 89591.63 90380.5 $/yr

TAC 169970.0 169987.04 $/yr

Table 8.5: Results and comparison with old optimal design

but it was an approximation to represent easily the all hyper-volume of uncertain-ties and hence to reduce the time to reach a satisfying minimum. Now it is possibleto verify how good was this approximation.

In figure 8.5 the number of points for each side of the hyper-rectangle of theprevious solution is progressively increased, in order to understand which couldbe the minimum number of points to approximate well the average OPEX of theprocess. It seems in this case that choosing 2 points for side, hence only vertices, isalready a good approximation for representing the whole process too; this meansconsidering 24 = 16 column simulations at each iteration. Nevertheless a betterchoice is to increase the number to 3, so considering also the medium point ofeach side, but this would mean 34 = 81 simulations of the column, about 5 timesmore in cpu-time at each iteration.

Increasing the number of points to at least 3 would lead to another advantage. Infact it would make sense to assign at each point a different probability accordingto a distribution, therefore multiplying each OPEX for a weight (and force the sumof weights to one). Therefore the reality would be better represented.


Figure 8.5: The points for each side are increased, in order to see how good was theapproximation of considering only vertices for evaluating the average OPEX.This is done for the solution of the previous optimization

9C O N C L U S I O N A N D F U T U R E D E V E L O P M E N T S

"Engineers believe 50% of what they hear, 75% of what they see and 100%of what comes out of a computer program."

Kister, H. Z.

Despite all the approximations done, a new methodology to consider flexibilityat the design stage was successfully applied to a case study and the result con-firmed the theoretical knowledge presented at the beginning, in Chapter 1. Noempirical factor, but a rigorous optimization at varying conditions has motivateda slight oversizing of the column. By the way, even if results are satisfactory, theyare only a first attempt and a further and more detailed design is required.

The column model (i.e. internals, utilities, costs,..) and the constrains can be de-scribed in a more detailed way, uncertainties can be increased in number (withdifferent weights) and typology (both discrete and boolean variables, where thesecond ones can define the existence of a species or an equipment). But then itis mandatory to reformulate the optimization problem and/or adopt a MINLPsolver, since it was showed before that this NLP one has several difficulties in han-dling even one non continuous variable. Moreover, it has to be kept in mind thatit is not possible to assure a global minimum and the decision maker has to besatisfied with a local one. More points, not only vertices, can be used to describein an accurate way the operational region and its probability distribution, if morecomputational power or cpu-time is available.

It would be easier to exploit a process simulator, maybe with ASCII files as vec-tors of information, to avoid modeling equipments already implemented in manycommercial softwares. Moreover, in a similar way it is possible to link the simula-tor to an external MINLP optimizer. This idea is inspired by the article of Corbetta,Grossman, and Manenti, 2015, where the C++ code is coupled with both PRO/II®and GAMS®. But a balance is needed, since a problem arises: the time used totransfer data from one software to the other and initialize the solution can influ-ence the overall time: it is noticeable that here with 4 uncertainties there are 16

vertices, hence 16 calls to the simulator software at each iteration plus one to theexternal optimizer. If each pass (with initialization) requires optimistically 0.1 [s],a total time of 0.1 · 17 ∼ 2[s] is required at each iteration only for passing data,with no calculation done. On the contrary, coding everything in one environment,

70 conclusion and future developments

if takes more in writing, it will speed up the overall convergence time.

These concepts hold true also for the methodologies explained in 6 and 7. Itwould be interesting to apply all these tree approach to more advanced configu-rations, such as biorefinery separations through a sequence of columns. However,since these are more complex than this conventional column with a quasi-idealmixture to separate, it is noticeable that the domain R has to be studied beforeevery approach is applied, to avoid infeasible points being included and flexibilityoverrated.

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Distillation Column: a Flexibility Study...Distillation Column: a Flexibility Study Supervisor: prof. flavio manenti Advisors: prof. ludovic montastruc prof. xavier joulia Master Graduation